Generally speaking, the more ACL2 knows, the slower it runs. That
is because the search space grows with the number of alternative
rules. Often, the system tries to apply rules that you have
forgotten were even there, if you knew about them in the first
place! ``Accumulated-persistence'' is a statistic (originally
developed for Nqthm) that helps you identify the rules that are
causing ACL2's search space to explode.

Accumulated persistence tracking can be turned on or off. It is
generally off. When on, the system runs about two times slower than
otherwise! But some useful numbers are collected. When it is
turned on, by

ACL2 !>(accumulated-persistence t)

an accumulation site is initialized and henceforth data about
which rules are being tried is accumulated into that site. That
accumulated data can be displayed with show-accumulated-persistence,
as described in detail below. When accumulated persistence is
turned off, with (accumulated-persistence nil), the accumulation
site is wiped out and the data in it is lost.

The ``accumulated persistence'' of a rune is the number of runes the
system has attempted to apply (since accumulated persistence was
last activated) while the given rune was being tried.

Consider a :rewrite rule named rune. For simplicity, let us imagine
that rune is tried only once in the period during which accumulated
persistence is being monitored. Recall that to apply a rewrite rule
we must match the left-hand side of the conclusion to some term we
are trying to rewrite, establish the hypotheses of rune by
rewriting, and, if successful, then rewrite the right-hand side of
the conclusion. We say rune is ``being tried'' from the time we
have matched its left-hand side to the time we have either abandoned
the attempt or finished rewriting its right-hand side. (By
``match'' we mean to include any loop-stopper requirement;
see loop-stopper.) During that period of time other rules
might be tried, e.g., to establish the hypotheses. The rules tried
while rune is being tried are ``billed'' to rune in the sense
that they are being considered here only because of the demands of
rune. Thus, if no other rules are tried during that period,
the accumulated persistence of rune is 1 -- we ``bill'' rune
once for its own application attempt. If, on the other hand, we
tried 10 rules on behalf of that application of rune, then
rune's accumulated persistence would be 11.

One way to envision accumulated persistence is to imagine that every
time a rune is tried it is pushed onto a stack. The rules tried on
behalf of a given application of a rune are thus pushed and popped
on the stack above that rune. A lot of work might be done on its
behalf -- the stack above the rune grows and shrinks repeatedly as
the search continues for a way to use the rune. All the while, the
rune itself ``persists'' in the stack, until we finish with the
attempt to apply it, at which time we pop it off. The accumulated
persistence of a rune is thus the number of stack frames built while
the given rune was on the stack.

Note that accumulated persistence is not concerned with whether the
attempt to apply a rune is successful. Each of the rules tried on
its behalf might have failed and the attempt to apply the rune might
have also failed. The ACL2 proof script would make no mention of
the rune or the rules tried on its behalf because they did not
contribute to the proof. But time was spent pursuing the possible
application of the rune and accumulated persistence is a measure of
that time.

A high accumulated persistence might come about in two extreme ways.
One is that the rule causes a great deal of work every time it is
tried. The other is that the rule is ``cheap'' but is tried very
often. We therefore keep track of the number of times each rule is
tried as well as its persistence. The ratio between the two is the
average amount of work done on behalf of the rule each time it is
tried.

When the accumulated persistence totals are displayed by the
function show-accumulated-persistence we sort them so that the most
expensive runes are shown first. We can sort according to one of
three keys:

:frames - the number of frames built on behalf of the rune
:tries - the number of times the rune was tried
:ratio - frames built per try

The key simply determines the order in which the information is
presented. If no argument is supplied to
show-accumulated-persistence, :frames is used.

Note that a rune with high accumulated persistence may not actually
be the ``culprit.'' For example, suppose rune1 is reported to have
a :ratio of 101, meaning that on the average a hundred and one
frames were built each time rune1 was tried. Suppose rune2 has a
:ratio of 100. It could be that the attempt to apply rune1 resulted
in the attempted application of rune2 and no other rune. Thus, in
some sense, rune1 is ``cheap'' and rune2 is the ``culprit'' even
though it costs less than rune1.

Users are encouraged to think about other meters we could install in
ACL2 to help diagnose performance problems.